Fast convergence of trust-regions for non-isolated minima via analysis of CG on indefinite matrices

Understand

Trust-region methods (TR) can converge quadratically to minima where the Hessian is positive definite.

• However, if the minima are not isolated, then the Hessian there cannot be positive definite.
• The weaker Polyak$\unicode{x2013}${\L}ojasiewicz (P{\L}) condition is compatible with non-isolated minima, and it is enough for many algorithms to preserve good local behavior.
• Yet, TR with an $\textit{exact}$ subproblem solver lacks even basic features such as a capture theorem under P{\L}.